A262709 Expansion of f(-x^4, -x^4) * f(-x, -x^5) in powers of x where f(, ) is Ramanujan's general theta function.

1, -1, 0, 0, -2, 1, 0, 0, 1, 2, 0, 0, -2, 0, 0, 0, 3, -2, 0, 0, -2, -3, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 2, -1, 0, 0, -2, 2, 0, 0, 1, 2, 0, 0, -4, 0, 0, 0, 0, -2, 0, 0, -2, 0, 0, 0, 3, 2, 0, 0, -2, 0, 0, 0, 2, -3, 0, 0, 0, 2, 0, 0, 4, 0, 0, 0, -2, 0, 0, 0, 2, -2

Offset

0,5

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-1/3) * eta(q) * eta(q^4)^2 * eta(q^6)^2 / (eta(q^2) * eta(q^3) * eta(q^8)) in powers of q.

Euler transform of period 24 sequence [ -1, 0, 0, -2, -1, -1, -1, -1, 0, 0, -1, -3, -1, 0, 0, -1, -1, -1, -1, -2, 0, 0, -1, -2, ...].

Example

G.f. = 1 - x - 2*x^4 + x^5 + x^8 + 2*x^9 - 2*x^12 + 3*x^16 - 2*x^17 + ...
G.f. = q - q^4 - 2*q^13 + q^16 + q^25 + 2*q^28 - 2*q^37 + 3*q^49 + ...

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^4] (EllipticTheta[ 4, 0, x^3] - EllipticTheta[ 4, 0, x^(1/3)]) / (2 x^(1/3)), {x, 0, n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A)^2 * eta(x^6 + A)^2 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^8 + A)), n))};

Crossrefs

Sequence in context: A369311 A263860 A051777 * A107628 A268389 A288969

Adjacent sequences: A262706 A262707 A262708 * A262710 A262711 A262712

Keyword

sign

Author

Michael Somos, Sep 28 2015

Status

approved